Theorem bdopf

Description: A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	bdopf	⊢ ( 𝑇 ∈ BndLinOp → 𝑇 : ℋ ⟶ ℋ )

Proof

Step	Hyp	Ref	Expression
1		bdopln	⊢ ( 𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp )
2		lnopf	⊢ ( 𝑇 ∈ LinOp → 𝑇 : ℋ ⟶ ℋ )
3	1 2	syl	⊢ ( 𝑇 ∈ BndLinOp → 𝑇 : ℋ ⟶ ℋ )